# Multi-Variable Calibration with iid Measurement Error

I now turn to whether measurement error in explanatory variables similar to those used by Lemmon, Roberts, and Zender (2008) is able to reproduce the leverage time series of both the actual- and residual-based portfolio sorts. I focus on profitability;

tangibility (as a measure of how tangible a firm’s collateral assets are); the market-to-book ratio (as a proxy for investment opportunities); and industry leverage (as a measure of industry-specific leverage targets, in lieu of an industry fixed effect). Firm size is excluded, since it is not stationary and thus would not conform to my setup of modelling the explanatory variables as AR(1) processes.

The estimation by simulated method of moments proceeds in a similar fashion to that in Section 3.2. In particular, the economy consists of simulated firms whose lever-age dynamics are governed by the following system of equations:

levit = β0(1 xit)0+ uit (26) ηit ∼ N (0, Ση).9 Leverage is determined in the cross-section by an intercept and the

9Vectors and matrices are denoted by bold letters.

four explanatory factors, which are all modelled as AR(1) processes. Firms differ in terms of the realization of a particular variable, but the coefficients in the model are the same for all firms. The true explanatory variable vector xit is latent; the observable xit is measured with error ηit: The explanatory variables are imperfect proxies for the true economic fundamentals driving leverage.

The covariance matrix of the innovations of the AR(1) processes Σ is diagonal, as is the covariance matrix of the measurement-error terms Ση:

Σ =

There are a total of 22 unknown parameters in this formulation: the intercepts, slopes, and error variances of the AR(1) process (12 parameters), the cross-sectional betas and the error variance σu2 (6 parameters), and the measurement-error variances (4 parame-ters).

To reduce the number of free parameters in the model, the unconditional means of the noisy explanatory variables are inferred directly from the data. This is possible since mismeasured and latent explanatory variables have the same mean: µx = E(xit) = E(xit + ηit) = E(xit) = µx. This allows me to express the intercepts of the latent AR(1) processes as functions of the empirical means of the respective variables and estimates of φ1, which is a free parameter matrix:

φ0 = (I4− φ1) µx (33)

= (I4− φ1) µx (34)

I4 denotes a 4 × 4 identity matrix. In fact, several other parameters could be inferred directly from the variance of leverage V ar(lev), and from the variance matrix of the noisy explanatory variables Σx.10 However, forcing the constraints that the model imposes on these parameters to hold exactly is too restrictive and results in a poor fit.

Instead, the variances are added as moment conditions, which results in the simulated values being close to the data values without the need to match them exactly.

10We could relate the variance matrix for the AR(1) innovations Σ to the variance of the noisy

3.3.1 Identification

To reduce the dimensionality of the parameter space, I calculate the intercepts for the autoregressive processes directly from the data via (34). This pares down the free structural parameters to a total of 18: the matrix φ1, which contains the slope coefficients for the explanatory variables, the innovation standard deviation matrix Σ, and the measurement-error variance matrix Ση. Furthermore, the parameter vector β, which governs the cross-sectional relationship between leverage and its determinants, along with the standard deviation of the cross-sectional residual σu, has to be estimated.

The structural parameters underlying the latent processes are obtained by matching simulated sample moments to data moments. Broadly speaking, the data moments con-sist of sample statistics for leverage and the explanatory variables, panel and time-series regression parameters, and the portfolio-leverage levels of the Lemmon, Roberts, and Zender (2008) portfolio sorts. Since I assume that the actual data on explanatory vari-ables are contaminated by measurement error, all data moments involving explanatory variables are mismeasured as well. In particular, I use the following moments:

i. The intercepts φ0 and slope coefficients φ1 for each explanatory variable (i.e., prof-itability, tangibility, market-to-book and industry leverage), which are obtained by regressing each observed mismeasured explanatory variable on its lagged value (8 moments):

xit = φ0+ φ1xit−1+ it (37) ii. The variance of each mismeasured explanatory variable σx2∗, and the variance of

leverage σ2lev (5 moments).

iii. The cross-sectional coefficients β from a regression of leverage on the noisy deter-minants (5 moments):

levit = β0(1 xit0) + uit where (38) β0 = (β0 βP rof βT ang βM B βIndLev ) (39) and xit is the vector of mismeasured explanatory variables.

factors Σx, and the variance of the regression residual σu2 to the variance of leverage V ar(lev) via:

Σ = I4− φ10φ1

 Σx

= I4− φ10

φ1 (Σx− Ση) (35)

σu2 = V ar(lev) − βΣxβ0 (36)

iv. The time series of portfolio-leverage levels after sorting on both actual and unex-pected leverage (80 moments in total). A time series consists of 20 portfolio-leverage levels for each “high-leverage” and “low-leverage” portfolio.

For both actual and simulated data, the moments are collected in vectors mact and msim, respectively. The structural parameters collected in the vector

Φ = (φ1 Σ Ση β σu) are found by minimizing the sum of the squared differences between actual moments and simulated moments:

min

Φ (mact− msim)0(mact− msim) (40)

This minimization makes the simulated moments as close to their actual counterparts as possible by picking the “best” structural parameter values.

3.3.2 Results

The estimated structural parameters of this procedure, along with their standard er-rors,11 are listed in Table 2. Table 3 presents a comparison of empirical data moments, their simulated counterparts based on mismeasured variables, and moments that are based on the estimated true latent parameters. Table 4 gives two estimates of the ra-tio of measurement noise to state noise for each simulated explanatory variable. The first estimate is the ratio of the measurement-error variance to the variance of the la-tent underlying variable, while the second estimate is the ratio of the measurement-error variance to the variance of the observed variable, which thus includes the measurement-error variance in the denominator. Finally, Figure 8 shows the portfolio sorts on actual and residual leverage, which are obtained with the estimated parameter values.

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For both the tangibility and industry-leverage ratios, the calibrated values of the latent processes are very close to the empirical data values. As measured by the AR(1)

11The standard errors are bootstrapped: First, all empirical moments are recalculated for subsets of the Compustat universe. I then estimate structural parameters for each of the subsamples. The standard errors are given by the standard deviations of the estimated structural parameters.

parameter and shown in Table 3, the estimated persistence for tangibility is 0.936 (empirical data value of 0.952), while it is 0.891 for industry leverage (empirical data value of 0.908). The estimated magnitude of the measurement-error standard deviation ση is small in both instances, and well below the standard deviation of the innovation σ in the respective AR(1) process (see Table 2). This results in a ratio of measurement-error variance to latent-variable variance ση22x of 0.021 for tangibility and 0.018 for industry leverage (see Table 4, column (1)). Similar values for the measurement-error ratio are obtained if the variance of the observed explanatory variable is used instead.

Consistent with the small magnitude of the estimated measurement error terms, the structural β-coefficients for both variables are close to their empirical counterparts (see Table 3).

Table 3 shows that latent profitability (φ1 = 0.832, see “Struc. Value” column) is more persistent than observed profitability (φ1 = 0.775). The depressed observed φ1 coefficient is caused by measurement error in observed profitability with an estimated standard deviation of ση = 0.105 (see Table 2), which also induces a slight downward bias in the cross-sectional β. Relative to tangibility and industry leverage, the mea-surement error ratios for profitability have increased to 0.090 and 0.083, respectively (see Table 4). These values are still low; for example, the latter implies that only 8.3%

of the variation in observed profitability is due to measurement error.

The most interesting result obtains for the market-to-book ratio. The latent AR(1) process has an estimated value of φ1 = 0.931, while the empirical process has a value of φ1 = 0.534 (see Table 3). The simulated φ1 value, obtained by regressing the simulated mismeasured market-to-book ratio on its lagged value, is 0.530, which is very close to the empirical estimate. The discrepancy between latent and observed φ1 is caused by a measurement-error standard deviation that is large when compared with that for the other variables. Its value is ση = 1.476, which exceeds the standard deviation of the innovation term in the AR(1) process σ = 0.603, as shown in Table 2. The resulting measurement-error ratio is ση2x2 = 0.802, which drops to ση22x = 0.445 if we use the variance of the observed market-to-book ratio in the denominator (see Table 4). This latter value implies that 44.5% of the observed variation in the market-to-book ratio is driven by noise. While this seems large, the market-to-book ratio as a proxy for investment opportunities can, ex ante, be expected to be noisy. Erickson and Whited (2006) state that “all observable measures or estimates of the true incentive to invest [...] are likely to contain measurement error.” Using a classical errors-in-variables model with the investment-to-capital ratio on the LHS and average q on the RHS, Erickson and Whited (2006) report that approximately 59% of the variation in book-value-based measures of Tobin’s q is driven by noise, and only 41% is driven by variation in the

true unobservable q. This is consistent with my results, where 55% of the variation in the market-to-book ratio is due to variation in true q.

My estimates of the structural parameters produce a variance in the observed market-to-book ratio of 4.895, which is equal to its empirical counterpart. Thus, the results are not driven by an unnaturally high total variance in the market-to-book ratio.

In the simulated cross-section, the true latent β-coefficient for the market-to-book ratio is -0.105, which is larger than the empirical value of -0.006 (Table 3). The simulated mismeasured observed value for βM B is -0.058. My results suggest that a market-to-book ratio, which is a poor proxy for true investment opportunities, plays an important role in the persistence of the residual-based portfolio sorts. Since an option to invest is riskier than the investment itself, firms with a high true q would optimally choose to carry lower amounts of leverage. However, this effect is obscured in the data owing to the high amount of measurement error inherent in the market-to-book ratio.

Overall, the estimation produces sensible parameter values, and the simulated mo-ments closely resemble their empirical data counterparts, as a comparison of the “Data Value” and “Sim. Value” columns in Table 3 reveals. Finally, Figure 8 shows the re-sults of the portfolio sorts. Using the estimated values of the structural parameters in Table 2 produces a close fit between empirical and simulated portfolio leverage time series, regardless of whether the sort is done on actual or residual leverage. While the simulated residual-based portfolios exhibit less dispersion than their empirical coun-terparts in years two to five, they track the empirical time series closely in the other time periods. This shows that low levels of measurement error in profitability, size and industry leverage, coupled with a larger, yet realistic, amount of measurement error inherent in using book-value-based proxies of Tobin’s q offers a potential explanation of the documented persistence in leverage portfolios.12

### 4 Conclusion

Persistence in residual-based leverage portfolios is a well-documented fact. While this persistence can result from the omission of either a firm fixed effect or time-varying variables, I show that it also arises when slow-moving explanatory variables in a lever-age regression are measured with error. Sorting firms into portfolios based on these regression residuals will exhibit similar portfolio-leverage persistence as sorting firms into portfolios based on actual leverage.

12In unreported results, model fit improves by allowing for a slight autocorrelation in the measurement-error terms themselves.

Being able to predict future leverage with the regression residuals implies that tar-get leverage is mismeasured. I find that if the leverage tartar-get is modelled as being determined by a single composite factor of a number of possible trade-off theory vari-ables, then the measurement-error variance of this latent factor needs to be 142% of its cross-sectional variance to reproduce the stylized empirical facts. This number is large, but is nonetheless a useful measure, since it can be interpreted as an aggregate estimate of how mismeasured the explanatory variables would need to be. However, even much lower amounts of measurement error still produce remarkably persistent residual-based portfolio sorts. Therefore, even if measurement error alone is not sufficient to fully ac-count for the persistence of leverage in the setting of regression-residual-based portfolio sorts, it is nonetheless likely to be an important contributor.

I also examine measurement error in several important explanatory variables, namely the firm’s profitability, the tangibility of its assets, the market-to-book ratio, and indus-try leverage. I find that low quantities of measurement error in profitability, tangibility and industry leverage, coupled with a measurement-error variance equal to about 80%

of the cross-sectional variation in the market-to-book ratio, produce a good fit of sim-ulated sample data moments to empirical moments. This level of measurement error in the market-to-book variable, which proxies for Tobin’s q, is consistent with other studies such as Erickson and Whited (2006), and suggests that unobserved investment opportunities play an important role in explaining leverage ratios, and, hence, in the persistence of the residual-based portfolio sorts.

The focus of this paper is on capital structure. However, portfolio sorts are also a popular tool to evaluate the returns from trading strategies, and to test asset pricing models. Measurement quality is an important consideration for the risk factors in these models, so my work also has implications for the asset pricing applications of portfolio sorts.